
\magnification = 2000    
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

%\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum


\nopagenumbers

\def\ni{\noindent}


%\phantom{.}\vskip-15pt

\vglue 20pt

\Title The Circle.  

% \centerline{ {\bf The Circle}
% {\footnote"*"{\verysmall 
%This file is from the 3D-XplorMath project.  
%Please see: \hfill\break \phantom{http://} http://3D-XplorMath.org/} } }

\medskip

\centerline{$x = aa\cos(t),\ y=aa\sin(t),\  0 \le t \le 2\pi$}
\LF
{\sc 3DXM - suggestion:} Select from the Action Menu {\it Show Generalized
Cycloid} and vary in the Settings Menu, entry:  {\it Set Parameters},
the (integer) ratio between the radius $aa$ and the rolling radius $hh$. \lf
The length of the drawing stick is $ii*$rolling radius.
\LF
The circle is the simplest and best known closed curve in the plane. The
default image shows the circle together with the theorem of Thales about
right angled triangles. Other properties of the circle are also known since
over 2000 years. In fact,  many of the plane curves that have 
individual names were already considered (and named) by the ancient
Greeks, and a large class of these can be obtained by rolling one {\it circle}
on the inside or the outside of some other {\it circle}. The Greeks were 
interested in rolling constructions because it was their main tool 
for describing the motions of the planets (Ptolemy). 
The following curves from the Plane Curve menu can be obtained 
by rolling constructions:
\bigskip
\goodbreak

\noindent
{\bf   Cycloid,  Ellipse,  Astroid,  Deltoid, Cardioid,  \hfil\break
Lima\c con, Nephroid, Epi- and Hypocycloids.}

\vskip0.5mm\noindent
Not all geometric properties of these curves follow easily
from their definition as rolling curve, but in some cases
the connection with complex functions (Conformal Category)
does.
\vskip0.5mm\noindent
{\bf Cycloids} arise by rolling a circle on a straight
line. The parametric equations code for such a cycloid is  
\vskip0.5mm\noindent
$ P.x := aa\cdot t - bb\sin(t)          \lf
P.y := aa - bb\sin(t) ,\   aa = bb .$
\vskip0.5mm\noindent
Cycloids have other cycloids of the same size as evolute (Action
Menu: ``Show Osculating Circles with Normals'').  This fact is
responsible for Huyghen's cycloid pendulum to have a period
independent of the amplitude of the oscillation.
\vskip0.5mm\noindent
{\bf Ellipses} are obtained if {\it inside} a circle of radius aa
another circle of radius  $r = hh = 0.5aa$ rolls and then traces
a curve with a radial stick of length  $R = ii\cdot r$.
The parametric equations for such an ellipse is  
\vskip0.5mm\noindent
$P.x := (R+r)\cos(t)          \lf
P.y := (R-r)\sin(t)    .$     
\vskip0.5mm\noindent
In the visualization of the complex map  $z \to z + 1/z$ in
Polar Coordinates the image of the circle of Radius R is such
an ellipse with $r = 1/R$.
\bigskip\medskip
\goodbreak\noindent
{\bf Astroids} are obtained if {\it inside} a circle of radius aa
another circle of radius  $r = hh = 0.25aa$ rolls and then traces
a curve with a radial stick of length $ R = ii\cdot r = r$.
Parametric equations for such Astroids are
\vskip0.5mm\noindent
$P.x := (aa-r)\cos(t) + R\cos(4t)         \lf
P.y := (aa-r)\sin(t) - R\sin(4t).$        
\vskip0.5mm\noindent
Astroids can also be obtained by rolling the {\it larger} circle
of radius r = hh = 0.75aa (put gg = 0 in this case).
Another geometric construction of the Astroids uses the fact
that the length of the segment of each tangent between the
x-axis and the y-axis has {\bf constant} length. --- Try $hh:= aa/3$
to obtain a {\bf Deltoid}.
\vskip0.5mm\noindent
{\bf Cardioids and Lima\c cons} are obtained if
{\it outside} a circle of radius $aa$
another circle of radius $ r = hh = - aa$ rolls and then traces
a curve with a radial stick of length $ R = ii\cdot r, ii = 1$ for
the Cardioids, $ii > 1$ for the Lima\c cons.
Their parametric equations are 
\vskip0.5mm\noindent
$P.x := (aa+r)\cos(t) + R\cos(2t)         \lf
P.y := (aa+r)\sin(t) + R\sin(2t)  .$
\vskip0.5mm\noindent
The Cardioids and Lima\c cons can also be obtained by rolling the
larger circle of radius $r = hh = + 2aa$; now $ ii < 1$ for the
Lima\c cons. Note that the fixed circle is {\it inside}
the larger rolling circle. 
The evolute of the Cardioid
(Action Menu: {\it Show Osculating Circles with Normals})
is a smaller Cardioid. The image of the unit circle
under the complex map $z \to w = (z^2 + 2z)$ is a Cardioid; images
of larger circles are Lima\c cons. Inverses $z \to 1/w(z) $ of Lima\c cons
are figure-eight shaped, one of them is a Lemniscate.
\vskip0.5mm\noindent
{\bf Nephroids} are generated by rolling a circle of one radius
outside of a second circle of twice the radius, as the program
demonstrates. With $R = 3r$ we thus have the parametrization 
\vskip0.5mm\noindent
$P.x := R\cos(t) + r\cos(3t)         \lf
P.y := R\sin(t) + r\sin(3t).$
\vskip0.5mm\noindent
As with Cardioids and Lima\c cons one can make the 
drawing stick shorter or longer: Choose in the Menu {\it Circle} and
set the parameters $aa = 1, hh = -0.5, ii = 1$ for the Nephroid and
$ii > 1$ for its looping relatives.
\vskip0.5mm\noindent
The complex map $ z \to z^3 + 3z$ maps the unit circle to such a
Nephroid. To see this, in the Conformal Map Category, select
$z \to z^{ee} + ee\cdot z$ from the Conformal Map Menu, then choose
Set Parameters from the Settings Menu and put $ee = 3$.
\goodbreak\noindent
Back in the Plane Curves Category, select Nephroid and then in the
Action Menu: {\it Show Osculating Circles with Normals}. The Normals
envelope a smaller Nephroid.
\vskip0.5mm\noindent
{\bf Archimedes' Angle Trisection.} A demo of this construction can be 
selected from the Action Menu.   


\noindent H.K.

\bye